Loading... ## 角度公式 ### **和差角公式** $$ \begin{aligned} &\cos(\alpha \pm \beta) = \cos\alpha \cos\beta \mp \sin\alpha \sin\beta, \\ &\sin(\alpha \pm \beta) = \sin\alpha \cos\beta \pm \cos\alpha \sin\beta, \\ &\tan(\alpha \pm \beta) = \frac{\tan\alpha \pm \tan\beta}{1 \mp \tan\alpha \tan\beta}. \end{aligned} $$ ### **二倍角公式** $$ \begin{aligned} &\cos(2\alpha) = \begin{cases} \cos^2\alpha - \sin^2\alpha \\ 2\cos^2\alpha - 1 \\ 1 - 2\sin^2\alpha \end{cases}\\ &\sin(2\alpha) = 2\sin\alpha \cos\alpha, \\ &\tan(2\alpha) = \frac{2\tan\alpha}{1 - \tan^2\alpha}. \end{aligned} $$ ### **和差化积公式** $$ \begin{aligned} &\sin\alpha + \sin\beta = 2\sin\left(\frac{\alpha+\beta}{2}\right)\cos\left(\frac{\alpha-\beta}{2}\right), \\ &\sin\alpha - \sin\beta = 2\cos\left(\frac{\alpha+\beta}{2}\right)\sin\left(\frac{\alpha-\beta}{2}\right), \\ &\cos\alpha + \cos\beta = 2\cos\left(\frac{\alpha+\beta}{2}\right)\cos\left(\frac{\alpha-\beta}{2}\right), \\ &\cos\alpha - \cos\beta = -2\sin\left(\frac{\alpha+\beta}{2}\right)\sin\left(\frac{\alpha-\beta}{2}\right). \end{aligned} $$ ### **积化和差公式** $$ \begin{aligned} &\sin\alpha \cos\beta = \frac{1}{2}\left[\sin(\alpha+\beta) + \sin(\alpha-\beta)\right], \\ &\cos\alpha \cos\beta = \frac{1}{2}\left[\cos(\alpha+\beta) + \cos(\alpha-\beta)\right], \\ &\sin\alpha \sin\beta = \frac{1}{2}\left[\cos(\alpha-\beta) - \cos(\alpha+\beta)\right]. \end{aligned} $$ ## 三角定理 ### 余弦定理（Law of Cosines） $$ \begin{aligned} a^2 &= b^2 + c^2 - 2bc \cos A, \\ b^2 &= a^2 + c^2 - 2ac \cos B, \\ c^2 &= a^2 + b^2 - 2ab \cos C. \end{aligned} $$ $$ \cos A = \frac{b^2 + c^2 - a^2}{2bc}, \quad \cos B = \frac{a^2 + c^2 - b^2}{2ac}, \quad \cos C = \frac{a^2 + b^2 - c^2}{2ab}. $$ ### 正弦定理 $$ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R, $$ 其中 $ R $ 为三角形的外接圆半径。 最后修改：2025 年 03 月 25 日 © 允许规范转载 打赏 赞赏作者 赞 如果觉得我的文章对你有用，请随意赞赏